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Physical Review B

Group-theoretical and resonating Hartree-Fock studies for spin fluctuations in two-dimensional Hubbard model on triangular lattice

Fumikazu Satoh, Masa-aki Ozaki, Takaaki Maruyama, Norikazu Tomita

Published Dec 7, 2011• 4 citations
Publisher

Abstract

Magnetic structures of the two-dimensional Hubbard model on a uniform triangular lattice are investigated from the viewpoint of quantum fluctuations. First, possible spin density wave (SDW) solutions in the unrestricted Hartree-Fock approximation are group theoretically classified for the ordering vectors of ${\mathbf{Q}}_{1}=({\mathbf{G}}_{a}+{\mathbf{G}}_{b})/4$, ${\mathbf{Q}}_{2}={C}_{3}^{+}{\mathbf{Q}}_{1}$, and ${\mathbf{Q}}_{3}={C}_{3}^{\ensuremath{-}}{\mathbf{Q}}_{1}$. Then, to include the electron correlation effects efficiently, we apply the resonating Hartree-Fock method. It is shown that the spin fluctuations in the metallic ground state at $U/t\ensuremath{\sim}4.5$ are reasonably described as hybridization of two different SDW states obtained by the group theoretical classification and their low-energy excited states. These fluctuations make the long-range magnetic order significantly reduced in the correlated ground state.

DOI
10.1103/physrevb.84.245101

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Magnetic structures of the two-dimensional Hubbard model on a uniform triangular lattice are investigated from the viewpoint of quantum fluctuations. First, possible spin density wave (SDW) solutions in the unrestricted Hartree-Fock approximation are group theoretically classified for the ordering vectors of Q1=(Ga+Gb)/4{\mathbf{Q}}_{1}=({\mathbf{G}}_{a}+{\mathbf{G}}_{b})/4Q1​=(Ga​+Gb​)/4, Q2=C3+Q1{\mathbf{Q}}_{2}={C}_{3}^{+}{\mathbf{Q}}_{1}Q2​=C3+​Q1​, and Q3=C3\ensuremath−Q1{\mathbf{Q}}_{3}={C}_{3}^{\ensuremath{-}}{\mathbf{Q}}_{1}Q3​=C3\ensuremath−​Q1​. Then, to include the electron correlation effects efficiently, we apply the resonating Hartree-Fock method. It is shown that the spin fluctuations in the metallic ground state at U/t\ensuremath∼4.5U/t\ensuremath{\sim}4.5U/t\ensuremath∼4.5 are reasonably described as hybridization of two different SDW states obtained by the group theoretical classification and their low-energy excited states. These fluctuations make the long-range magnetic order significantly reduced in the correlated ground state.
Group theoretically possible spin density waves (SDWs) of the Hubbard model on a triangular lattice are found and classified for ordering vectors q of K and M in the Brillouin zone by finding the maximal subgroups of the symmetry group of the Hamiltonian. It is found that classifications of SDWs with these ordering vectors are reduced to classifications of SDWs on molecules with D 3 and T d symmetries, respectively. From this reduction the authors found two axial-type ASDWs (ASDW) and one plane-type SDW (PSDW) for point K and three ASDWs, two PSDWs and one cubic- (three-dimensional-) type SDW (CSDW) for point M. The standard mean-field Hamiltonians and their invariance groups are listed for each class.
The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree-Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping, h. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is 2/h. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolve into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.
Systems of fermions described by the three-dimensional (3D) repulsive Hubbard model on a cubic lattice have recently attracted considerable attention due to their possible experimental realization via cold atoms in an optical lattice. Analytical and numerical results are limited away from half-filling. We study the ground state of the doped system from weak to intermediate interaction strengths within the generalized Hartree-Fock approximation. The exact solution to the self-consistent-field equations in the thermodynamic limit is obtained and the ground state is shown to exhibit antiferromagnetic order and incommensurate spin-density waves (SDW). At low interaction strengths, the SDW has unidirectional character with a leading wavevector along the [100]-direction, and the system is metallic. As the interaction increases, the system undergoes a simultaneous structural and metal-to-insulator transition to a unidirectional SDW state along the [111]-direction, with a different wavelength. We systematically determine the real and momentum space properties of these states. The crossover from 3D to two dimensions (2D) is then studied by varying the inter-plane hopping amplitude, which can be experimentally realized by tuning the distance between a stack of square-lattice layers. Detailed comparisons are made between the exact numerical results and predictions from the pairing model, a variational ansatz based on the pairing of spins in the vicinity of the Fermi surface. Most of the numerical results can be understood quantitatively from the ansatz, which provides a simple picture for the nature of the SDW states.
The electronic structure of the one dimensional Hubbard model is calculated by the resonating Hartree-Fock (HF) method. Solitons in the spin density wave (SDW) order parameter (OP) make translational bands with a lot of high harmonics. Their effective masses are about 0.1 of the electron mass, so that they are potentially good charge or spin carriers. We consider diradical and zwitterionic breathers, which consist of a neutral and charged soliton-antisoliton pair, respectively, making quantum translational and breathing motions, as local quantum fluctuations in the SDW in the half filled case. The stabilization of the ground state by the resonationg breathers amounts to 76.5% ( U =3) and 70.1% ( U =4) of the correlation energy. This is better than the result of the Gutzwiller method. The fluctuations by breathers explain the rapid decrease of the spin correlation function in short distances and give also low energy excited states with the symmetries and energetic ordering 3 B < 1 A < 1 B that are characteristic in polyenes.
We use unrestricted Hartree-Fock, density matrix renormalization group, and variational projected entangled-pair state calculations to investigate the ground-state phase diagram of the triangular-lattice Hubbard model at “half doping” relative to single occupancy, i.e., at fillings of ( 1 ± 1 2 ) electrons per site. The electron-doped case has a nested Fermi surface in the noninteracting limit, and hence a weak-coupling instability toward density-wave orders whose wave vectors are determined by Fermi-surface nesting conditions. We find that at moderate-to-strong interaction strengths, other spatially modulated orders arise, with wave vectors distinct from the nesting vectors. In particular, we identify a series of closely competing, itinerant long-wavelength magnetically ordered states, yielding to uniform ferromagnetic order at the largest interaction strengths. For half-hole doping and a similar range of interaction strengths, our data indicate that magnetic orders are most likely absent.